This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A219023 #20 Sep 23 2023 03:40:01
%S A219023 0,0,0,0,0,0,0,1,1,1,0,2,0,1,1,0,0,2,1,0,2,0,0,0,2,1,1,0,2,1,0,2,3,0,
%T A219023 2,2,0,1,4,1,2,1,0,0,3,1,1,3,0,0,1,2,1,1,1,1,0,0,2,3,1,0,3,1,2,1,0,1,
%U A219023 4,0,1,2,0,2,3,0,0,4,0,2,2,0,1,3,2,1,4,1,1,3,3,2,3,1,2,1,0,2,4,2
%N A219023 Number of primes p<n such that n^2-n+p and n^2+n-p are both prime.
%C A219023 Conjecture: a(n)>0 for all n>2732.
%C A219023 We have verified this conjecture for n up to 1.4*10^7. Note that the conjecture is stronger than Oppermann's conjecture which states that for any integer n>1 both of the two intervals (n^2-n,n^2) and (n^2,n^2+n) contain primes.
%C A219023 Zhi-Wei Sun also made the following conjectures: For n>3512 there is a prime p in (n,2n) such that both n^2-n+p and n^2+n-p are prime. For n>1828 there is a prime p<n such that both n^2-n-p and n^2+n+p are prime. For n>4517 there is a prime in (n,2n) such that both n^2-n-p and n^2+n+p are prime.
%H A219023 Zhi-Wei Sun, <a href="/A219023/b219023.txt">Table of n, a(n) for n = 1..20000</a>
%H A219023 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588 [math.NT], 2012-2017.
%H A219023 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann's_conjecture">Oppermann's conjecture</a>
%e A219023 a(12)=2 since the 5 and 7 are the only primes p<12 with 12^2-12+p and 12^2+12-p both prime.
%t A219023 a[n_]:=a[n]=Sum[If[PrimeQ[n^2-n+Prime[k]]==True&&PrimeQ[n^2+n-Prime[k]]==True,1,0],{k,1,PrimePi[n-1]}]
%t A219023 Do[Print[n," ",a[n]],{n,1,20000}]
%t A219023 Table[Total[Table[If[AllTrue[{k^2-k+p,k^2+k-p},PrimeQ],1,0],{p,Prime[ Range[ PrimePi[k]]]}]],{k,100}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 23 2017 *)
%o A219023 (PARI) A219023(n)={my(c=0,nm=n^2-n,np=n^2+n); forprime(p=1,n-1,isprime(np-p) && isprime(nm+p) && c++); c} \\ - _M. F. Hasler_, Nov 11 2012
%Y A219023 Cf. A000040.
%K A219023 nonn
%O A219023 1,12
%A A219023 _Zhi-Wei Sun_, Nov 10 2012